Treatment vs. effect contrasts

Hi Wolfgang and Michel, here is a simple example to show the differences between treatment and effect contrasts.

Example dataset

Create a random dataset with criteria y, predictors x1, x2 and gender.

All variables are correlated.

set.seed(1234)
n <- 10000
gender <- rep(0:1, each = n/2)
y <- sample(0:10, n, replace = TRUE) + gender * sample(0:10, n, replace = TRUE)
x1 <- sample(0:10, n, replace = TRUE) + gender * y
x2 <- x1 + sample(0:10, n, replace = TRUE) + gender * y
inter <- x1 > median(x1) & x2 > median(x2) & gender == 1
y[inter] <- y[inter] + sample(0:10, sum(inter), replace = TRUE)

dat <- data.frame(y = y, x1 = x1, x2 = x2, gender = gender)

Descriptives

psych::corr.test(dat)

Call:psych::corr.test(x = dat)
Correlation matrix 
          y   x1   x2 gender
y      1.00 0.79 0.83   0.66
x1     0.79 1.00 0.94   0.74
x2     0.83 0.94 1.00   0.79
gender 0.66 0.74 0.79   1.00
Sample Size 
[1] 10000
Probability values (Entries above the diagonal are adjusted for multiple tests.) 
       y x1 x2 gender
y      0  0  0      0
x1     0  0  0      0
x2     0  0  0      0
gender 0  0  0      0

 To see confidence intervals of the correlations, print with the short=FALSE option

psych::describe(dat)

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
y	1	10000	9.5326	6.768879	8.0	9.012750	7.4130	0	30	30	0.6222269	-0.5002773	0.0676888
x1	2	10000	10.0063	6.672279	9.0	9.585875	7.4130	0	30	30	0.5162774	-0.5350028	0.0667228
x2	3	10000	19.9998	12.575874	16.0	19.046000	11.8608	0	58	58	0.5910003	-0.6639372	0.1257587
gender	4	10000	0.5000	0.500025	0.5	0.500000	0.7413	0	1	1	0.0000000	-2.0002000	0.0050003

Contrasts

The left part of the table is with gender as treatment contrast (0 vs. 1) and the right part with gender as effect contrast (-1 vs. 1)

# Gender has values 0 vs. 1 (treatment contrast)
fit1 <- lm(y ~ gender * x1 * x2, data = dat)

# Gender hast -1 vs. 1 (effect contrast)
dat$gender <- car::recode(dat$gender, "0 = -1; 1 = 1")

fit2 <- lm(y ~ gender * x1 * x2, data = dat)

sjPlot::tab_model(fit1, fit2, show.se = TRUE, show.ci = FALSE, col.order = c("est", "se", "std.est", "p"), digits = 4)

	y			y
Predictors	Estimates	std. Error	p	Estimates	std. Error	p
(Intercept)	5.1849	0.1880	<0.001	-0.7110	0.1882	<0.001
gender	-11.7920	0.3764	<0.001	-5.8960	0.1882	<0.001
x1	0.0226	0.0420	0.590	0.2945	0.0259	<0.001
x2	-0.0066	0.0239	0.781	0.2983	0.0144	<0.001
gender × x1	0.5438	0.0519	<0.001	0.2719	0.0259	<0.001
gender × x2	0.6098	0.0288	<0.001	0.3049	0.0144	<0.001
x1 × x2	-0.0026	0.0036	0.479	-0.0072	0.0018	<0.001
(gender × x1) × x2	-0.0093	0.0037	0.011	-0.0047	0.0018	0.011
Observations	10000			10000
R2 / R2 adjusted	0.747 / 0.747			0.747 / 0.747